A Positive Proportion of the Reduced D'Arcais Polynomials is not Hurwitz

Heretofore, the second and third author conjectured that the D’Arcais polynomials, related to the coefficients of the powers of the Dedekind $\eta$-function, are Hurwitz polynomials except for a root at the origin. We show that this does in fact not hold for a positive proportion of all natural numbers.